PreprintCC BY 4.0arXiv
Exceptional pretzel knots are not algebraically slice
Abstract
We consider the three-strand pretzel knot Pa=P(a,−a−2,−(a+1)2/2) and prove that, when a≡1pmod4, its Alexander polynomial fails the Fox--Milnor condition. The same remains true after adding any number of opposite odd parameter pairs q,−q and reordering the parameters. Combined with work of Kim--Lee--Song, this gives the same conclusion for every odd a≥3. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots.
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BibTeX
@misc{saurabh2026exceptional,
title = {{Exceptional pretzel knots are not algebraically slice}},
author = {Suman Saurabh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34522},
eprint = {2609.34522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exceptional pretzel knots are not algebraically slice
AU - Suman Saurabh
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.34522
UR - https://arxiv.org/abs/2609.34522
AB - We consider the three-strand pretzel knot P_a=P(a,−a−2,−(a+1)^2/2) and prove that, when a≡1pmod4, its Alexander polynomial fails the Fox--Milnor condition. The same remains true after adding any number of opposite odd parameter pairs q,−q and reordering the parameters. Combined with work of Kim--Lee--Song, this gives the same conclusion for every odd a≥3. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots.
ER -